Continuous dependence for McKean-Vlasov SDEs under distribution-dependent Lyapunov conditions

TL;DR

This paper investigates how solutions and invariant measures of McKean-Vlasov SDEs depend continuously on initial conditions and parameters, highlighting differences from classical SDEs due to measure distance mismatches.

Contribution

It establishes continuous dependence results for McKean-Vlasov SDEs under distribution-dependent Lyapunov conditions, addressing convergence issues related to measure distances.

Findings

Solutions do not converge in probability despite initial convergence.

Invariant measures depend continuously on initial conditions.

Examples illustrate the theoretical results.

Abstract

In this paper, we consider the continuous dependence on initial values and parameters of solutions as well as invariant measures for McKean-Vlasov SDEs under distribution-dependent Lyapunov conditions. In contrast to the classical SDEs, the solutions for McKean-Vlasov SDEs do not converge in probability although the initial values converge in probability, which is due to the mismatch of the distances between measures. Finally, we give some examples to illustrate our theoretical results.